(a) 1 Show that Cπ/2 S * de ƒ (0) = √ √„** do [ƒ (0) + ƒ(0)] where f() is any function of 0. (b) Show that the total flux of particles along the z-axis passing the area element is F₂ π/2 == 1 -v de sin # cos Ꮎ dr P(r) [n(r cos 0) - n(-r cos 0)] where is the mean particle speed. Hint: Use the result of part (a) with f(0) = sin cos fdr Pr(r)n(r cos 0) and employ the identities sin(7-0) = sin and cos( 0) = = cos 0. -

One-dimensional harmonic oscillators in equilibrium with a heat bath (a) Calculate the specific heat of the one-dimensional harmonic oscillator as a function of temperature (b) Plot the T -dependence of the mean energy per particle E/N and the specific heat c. Show that E/N → kT at high temperatures for which kT > hw In this limit the energy kT is large in comparison to hw , the separation between energy levels. Hint: expand the exponential function 1 ē = ħw + eBhw

Plz solve in 30 min I vill definitely upvote and positive rating

Problem 2) Consider the following Maxwell Boltzmann distribution of molecular speeds: P(v) = 4( m 27kBT. mp² v²e 2kgT To calculate average values for say f(v) (function of v) one just integrates f(v) with P(v)dv from zero to infinity = P(v)f(v)dv, where signifies average of f(v). Of course, the distribution should be normalized: P(v)dv=1, (is a requirement for any probability distribution). a) Check the last equation. b) Calculate the average of v. c) Calculate the average of v². d) Calculate from c) the RMS value of the speed. e) Calculate the most probable value of v. f) Square the results of b, d and e and rank them from smallest to the largest value.